Average Error: 6.4 → 0.7
Time: 5.0s
Precision: 64
\[\frac{x \cdot y}{z}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \cdot y \le -1.394495778794157817173263708359068376665 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -2.091115834215142392182824625708863732465 \cdot 10^{-215}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.107151871760683778945393640864932427286 \cdot 10^{-309}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 8.013896806549786117243748175613103383758 \cdot 10^{151}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]
\frac{x \cdot y}{z}
↓
\begin{array}{l}
\mathbf{if}\;x \cdot y \le -1.394495778794157817173263708359068376665 \cdot 10^{144}:\\
\;\;\;\;\frac{x}{z} \cdot y\\

\mathbf{elif}\;x \cdot y \le -2.091115834215142392182824625708863732465 \cdot 10^{-215}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{elif}\;x \cdot y \le 1.107151871760683778945393640864932427286 \cdot 10^{-309}:\\
\;\;\;\;\frac{x}{\frac{z}{y}}\\

\mathbf{elif}\;x \cdot y \le 8.013896806549786117243748175613103383758 \cdot 10^{151}:\\
\;\;\;\;\frac{x \cdot y}{z}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{z} \cdot y\\

\end{array}
double f(double x, double y, double z) {
        double r792132 = x;
        double r792133 = y;
        double r792134 = r792132 * r792133;
        double r792135 = z;
        double r792136 = r792134 / r792135;
        return r792136;
}
↓
double f(double x, double y, double z) {
        double r792137 = x;
        double r792138 = y;
        double r792139 = r792137 * r792138;
        double r792140 = -1.3944957787941578e+144;
        bool r792141 = r792139 <= r792140;
        double r792142 = z;
        double r792143 = r792137 / r792142;
        double r792144 = r792143 * r792138;
        double r792145 = -2.0911158342151424e-215;
        bool r792146 = r792139 <= r792145;
        double r792147 = r792139 / r792142;
        double r792148 = 1.107151871760684e-309;
        bool r792149 = r792139 <= r792148;
        double r792150 = r792142 / r792138;
        double r792151 = r792137 / r792150;
        double r792152 = 8.013896806549786e+151;
        bool r792153 = r792139 <= r792152;
        double r792154 = r792153 ? r792147 : r792144;
        double r792155 = r792149 ? r792151 : r792154;
        double r792156 = r792146 ? r792147 : r792155;
        double r792157 = r792141 ? r792144 : r792156;
        return r792157;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.4
Target6.3
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;z \lt -4.262230790519428958560619200129306371776 \cdot 10^{-138}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \lt 1.704213066065047207696571404603247573308 \cdot 10^{-164}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (* x y) < -1.3944957787941578e+144 or 8.013896806549786e+151 < (* x y)

    1. Initial program 19.0

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*2.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
    4. Using strategy rm

    5. Applied associate-/r/3.0

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y}\]

    if -1.3944957787941578e+144 < (* x y) < -2.0911158342151424e-215 or 1.107151871760684e-309 < (* x y) < 8.013896806549786e+151

    1. Initial program 0.2

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*9.4

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]

    4. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]

    if -2.0911158342151424e-215 < (* x y) < 1.107151871760684e-309

    1. Initial program 14.4

      \[\frac{x \cdot y}{z}\]
    2. Using strategy rm

    3. Applied associate-/l*0.3

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{y}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \le -1.394495778794157817173263708359068376665 \cdot 10^{144}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \cdot y \le -2.091115834215142392182824625708863732465 \cdot 10^{-215}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;x \cdot y \le 1.107151871760683778945393640864932427286 \cdot 10^{-309}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;x \cdot y \le 8.013896806549786117243748175613103383758 \cdot 10^{151}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.Solve.Tridiagonal:solveCyclicTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -4.262230790519429e-138) (/ (* x y) z) (if (< z 1.7042130660650472e-164) (/ x (/ z y)) (* (/ x z) y)))

  (/ (* x y) z))